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%I #7 Mar 24 2019 22:58:45
%S 1,1,3,1,15,15,1,1,43,161,105,10,1,96,855,1680,855,96,1,1,185,3191,
%T 13387,17655,7623,945,21,1,323,9570,72254,188188,188188,72254,9570,
%U 323,1,1,525,24675,302359,1345605,2499861,2036125,715725,99414,4410,36,1,808
%N Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2*n-1) * Sum_{s>=0} A001263(n+2*s,2*s+1)*x^s in row n >= 1 and column k >= 0.
%C Row sums are 1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, ... (see A151403, A052704).
%C The sequence is the Narayana number analog of A034839.
%e 1;
%e 1, 3;
%e 1, 15, 15, 1;
%e 1, 43, 161, 105, 10;
%e 1, 96, 855, 1680, 855, 96, 1;
%e 1, 185, 3191, 13387, 17655, 7623, 945, 21;
%e 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1;
%e 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36;
%e 1, 808, 56896, 1055320, 7329975, 22338816, 32152848, 22338816, 7329975, 1055320, 56896, 808, 1;
%p A001263 := proc(n,k) if n <=0 or k <=0 then 0 ; elif k > n then 0 ; else binomial(n-1,k-1)*binomial(n,k-1)/k ; end if; end proc:
%p A178657 := proc(n,k) (1-x)^(2*n-1)*add(A001263(n+2*l,2*l+1)*x^l,l=0..20) ; expand(%) ; coeftayl(%,x=0,k) ; end proc: # _R. J. Mathar_, Aug 30 2011
%t p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[2*k + n, 2*k] Binomial[ 2*k + n, 1 + 2*k]/(2*k + n))*x^k, {k, 0, Infinity}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 2, 10}];
%t Flatten[%]
%Y Cf. A001263, A034839.
%K nonn,tabf
%O 1,3
%A _Roger L. Bagula_, Jun 01 2010
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